In this work, we provide a simple derivation of the same upper bound on the minimum error probability as a function of the QFI. The same method also provides a loose lower bound. Using these, we investigate possible advantages of using definite-photon number entangled states of the form \(\sum _{n=0}^{N} a_n |N-n,n\rangle _{SI}\) , which we will call \(N\) -photon entangled (\(N\) PE) states. These states have been shown to be useful for quantum estimation purposes [1]}. Possible advantages of using \(N\) PE states in QI were first investigated in Ref. [2]}, where an optimized \(N\) PE state was shown to outperform the coherent state under repetitive local measurement strategies. The optimum \(N\) PE state, for a given value of \(N\) , was found by numerically maximizing the QFI. Furthermore, by considering a specific non-optimal scheme, a performance measure (signal-to-noise ratio) was shown to increase with a thermal photon number for large enough values of the latter. Here, we show that, contrary to the findings in Ref. [2]}, the QFIs for the optimized \(N\) PE states are slightly lower than those of the coherent states when the signal energy is constrained to be equal. By performing an exact calculation, we also show that the signal-to-noise ratio decreases monotonically as thermal noise is increased.
